We recall that \(L^2(pw)=E^2(p,w)\)  is separable. If \(\widetilde{\mathcal {L}}\)  is compact, as it is also self-adjoint and positive, (A1) and (A2) follow from Theorems VI.15 and VI.16 in [1]}. Theorem 4.6 in chapter 2 of [2]} tells us that, under (A1)-(A2), \(\widetilde{\mathcal {L}}\)  is compact.
The same Theorem ensures that \(\left\Vert \widetilde{\mathcal {L}}\right\Vert _{E^2(p,w)\rightarrow E^2(p,w)}\)  is the largest eigenvalue of \(\widetilde{\mathcal {L}}\) . So, we get (A3) from Proposition REF .
